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ANNALS OF THE

UNIVERSITY OF CRAIOVA

Series: AUTOMATION, COMPUTERS, ELECTRONICS and MECHATRONICS

Vol. 12 (39), No. 1, 2015 ISSN 1841-0626

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EDITURA UNIVERSITARIA

ANNALS OF THE UNIVERSITY OF CRAIOVA Series: AUTOMATION, COMPUTERS, ELECTRONICS AND MECHATRONICS

Vol. 12 (39), No. 1, 2015 ISSN 1841-0626

Note: The “Automation, Computers, Electronics and Mechatronics Series” emerged from “Electrical Engineering Series” (ISSN 1223-530X) in 2004.

Honorary Editor:

Vladimir RĂSVAN – University of Craiova, Romania

Editor-in-Chief: Emil PETRE – University of Craiova, Romania

Associate Editors-in-Chief: Marius BREZOVAN – University of Craiova, Romania Dorian COJOCARU – University of Craiova, Romania Dan SELIȘTEANU – University of Craiova, Romania

Editorial Board:

Costin BĂDICĂ – University of Craiova, Romania Andrzej BARTOSZEWICZ – Institute of Automatic Control, Technical University of Lodz, Poland Nicu BÎZDOACĂ – University of Craiova, Romania Eugen BOBAŞU – University of Craiova, Romania David CAMACHO – Universidad Autonoma de Madrid, Spain Kazimierz CHOROS – Wroclaw University of Technology, Poland Ileana HAMBURG – Institute for Work and Technology, FH Gelsenkirchen, Germany Mirjana IVANOVIC – University of Novi Sad, Serbia Mircea IVĂNESCU – University of Craiova, Romania Vladimir KHARITONOV – University of St. Petersburg, Russia Peter KOPACEK – Institute of Handling Device and Robotics, Vienna University of

Technology, Austria Rogelio LOZANO – CNRS – HEUDIASYC, France Dan Bogdan MARGHITU – Auburn University, Alabama, USA Marius MARIAN – University of Craiova, Romania Mihai MOCANU – University of Craiova, Romania Sabine MONDIÉ – CINVESTAV (Department of Automatic Control), Mexico Ileana NICOLAE – University of Craiova, Romania Silviu NICULESCU – CNRS – SUPELEC (L2S), France Mircea NIŢULESCU – University of Craiova, Romania Sorin OLARU – CNRS – SUPELEC (Automatic Control Department), France Octavian PASTRAVANU – “Gheorghe Asachi” Technical University of Iasi, Romania Dan PITICĂ – Technical University of Cluj-Napoca, Romania Dan POPESCU – University of Craiova, Romania Radu-Emil PRECUP – “Politehnica” University of Timisoara, Romania Dorina PURCARU – University of Craiova, Romania Dan STOIANOVICI – Johns Hopkins University, Baltimore, Maryland, USA Sihem TEBBANI – CNRS – SUPELEC (Automatic Control Department), France

Editorial Secretary

Elvira POPESCU – University of Craiova, Romania

Associate Editorial Secretary Monica-Gabriela ROMAN – University of Craiova, Romania

Address for correspondence: Emil PETRE University of Craiova, Faculty of Automation, Computers and Electronics Al.I. Cuza Street, No. 13, RO-200585, Craiova, Romania Phone: +40-251-438198, Fax: +40-251-438198 Email: [email protected]

We exchange publications with similar institutions from country and from abroad

CONTENTS

Eugen IANCU, Sergiu IVANOV, Eugen BOBAȘU, Emil PETRE: Method for Fault Detection

1

Camelia MAICAN, Gabriela CĂNURECI: Fault Detection in Educational Kit Festo

7

Sergiu IVANOV, Dan SELIŞTEANU, Virginia IVANOV, Dorin ŞENDRESCU: Compact Dynamic Model of the Brushless DC motor

13

Emil PETRE: Adaptive and Predictive Control Algorithms for a Microalgae Process

17

Eugen IANCU, Emil PETRE: Method for Anticipative Control of Bioprocess 24

Bogdan POPA, Dan POPESCU: Improving of the Backtracking Algorithm using different strategy for solving the 2-d problems

29

Adrian RUNCEANU: An implementation in BlueJ used in teaching object-oriented programming

34

Călin CONSTANTINOV, Mihai MOCANU, Cosmin POTERAȘ: Running Complex Queries on a Graph Database: A Performance Evaluation of Neo4j

38

Author Index 45

ANNALS OF THE UNIVERSITY OF CRAIOVA Series: Automation, Computers, Electronics and Mechatronics, Vol. 12 (39), No. 1, 2015

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1

Method for Fault Detection

Eugen Iancu*, Sergiu Ivanov**, Eugen Bobașu*, Emil Petre*

*Department of Automation and Electronic, University of Craiova, 107 Decebal Street, RO-200440 Craiova, Romania (e-mail: [email protected], http://www.ace.ucv.ro)

**Department of Electromechanics, Environment and Industrial Informatics, University of Craiova, 107 Decebal Street, RO-200440 Craiova

Abstract: This study shows a method for analytical fault detection that can be applied to monitor an electric motor brushless DC. The fault detection and the isolation (FDI) problem is an inherently complex one and for this reason the immediate goals is to preserve the stability of the process and, if is possible, to control the process in a slightly degraded manner. The authors propose a practical method to detect the presence of failures of sensors using a prediction solution. It was used for this purpose a mathematical model of BLDC motor and single exponential smoothing. Also is proposed a structure to detect the presence of failures.

Keywords: Brushless DC motor, fault detection and isolation, analytical redundancy, single exponential smoothing.

1. INTRODUCTION

Changes (faults) can make the system unsafe and less reliable. Productivity of the automatic system can degrade because changes can impose performance limitations on the system and may also require frequent system shut downs for its maintenance. In order to avoid production deteriorations or damage to machines and humans, variations have to be detected as quickly as possible and decisions that stop the propagation of their effects have to be made (Nguang et al., 2005).

The necessity to obtain a diagnostic with good performances, without installing a lot of redundant and dedicated expensive equipment, forces the diagnostic tools to develop the techniques available to processing all the information that are "hidden" in the technological process. In fact which the reality of industrial systems can offer to the engineer charged to implement the monitoring functions, is usually very inadequate: poor models available, lack of redundancy, insufficient number of measures, noise on the acquired data, unmodeled disturbances, etc.

The problem of reliable fault diagnosis in dynamic processes has received great attention and a wide variety of robust approaches has been proposed and developed. Analytical redundancy–based methods have been developed to diagnose faults in linear, time invariant, dynamic systems and a wide variety of model–based approaches has been proposed (Chen and Patton, 1999).

All failure detection methods exploit redundant data, which are obtained either directly, when two or more sensors are available for measurement of a process variable, or analytically, when a process variable is estimated using the mathematical process model. These redundancy relationships may then be exploited to

generate residual signals. Under normal operating conditions these residuals are "small" in an appropriate sense and yet display distinct patterns when failures occur.

Fig. 1. The structure of the analytical diagnostic (Iancu and Vinatoru, 2005).

Process

Data acquisition

Knowledge base

Mathematical model

Residual generation

Logical analysis

u y

Fault detection - fault location - cause of fault - fault size - fault time

Evaluation of consequences

ANNALS OF THE UNIVERSITY OF CRAIOVA

Series: Automation, Computers, Electronics and Mechatronics, Vol. 12 (39), No. 1, 2015 ____________________________________________________________________________________________________________

2

The failure diagnosis process consists in three stages (Fig. 1):

Modeling of process Residual generation Residual analysis.

The residuals must be carefully examinated to determine the presence of failures (detection) and which system components have failed (isolation). In practice it is often difficult to fulfil the demands of the method for the complex diagnosed plant. Robust methods of diagnosis are therefore required, in the face of existing measurement uncertainty, disturbances and incomplete knowledge (Patton, 1994). In such cases an integrated approach using quantitative and qualitative models in diagnostic expert systems gives a good solution.

2. MATHEMATICAL MODEL OF BRUSHLESS DIRECT CURRENT MOTOR

Brushless DC motor (BLDC) is a electric motor, type synchronous, usually three-phase, having rotor with permanent magnets and stator built with concentrated or uniformly distributed windings. Specific electronic circuit is accomplished by switching semiconductor elements of the three-phase power inverter that is synchronized with the rotor position. Rotor position must be known at specific angles to align with the applied electric voltage. Usually, in order to obtain information on the rotor possition are using three Hall effect sensors, encapsulated in stator. The mathematical model of the drive can be made using the principle ilustrated in the equivalent scheme represented in Fig.2.

Ud C

id

T1 T3 T5

T4 T6 T2

D1 D3 D5

D4 D6 D2

ia ib

ic

Rs Ls – Lm ea

Rs Ls – Lm eb

Rs Ls – Lm ec

ua

ub

uc

Fig. 2. Schematic diagram of the drive with brushless DC motor.

The equations of the phase voltages, are (Kennedy and Eberhart, 1995; Fedák et al., 2012):

a b ca s a s m m a

di di diu R i L L L edt dt dt

= + + + + , (1)

a b cb s b m s m b


= + + + + , (2)

a b cc s c m m s c


= + + + + , (3)

where: ua, ub, uc – instantaneous values of the phase voltages;

ea, eb, ec – the instantaneous electric phase voltages; ia, ib, ic – the instantaneous phase currents; Rs, Ls – phase stator resistance and inductance, assumed the same on all phases; Lm – the mutual inductance.

Given the star connection of the stator windings, provided

0a b ci i i+ + = (4)

allows writing equations (1) - (3) as:

( ) aa s a s m adiu R i L L edt

= + − + ; (5)

( ) bb s b s m bdiu R i L L edt

= + − + ; (6)

( ) cc s c s m cdiu R i L L edt

= + − + . (7)

Electric phase voltages depend by the position of the rotor to form a balanced three-phase system, the waveform imposed by the trapezoidal rule (Fig. 3). Thus, they can be expressed as (Fedák et al., 2012; Prasad et al., 2012):

( )a e ee K f= ω⋅ ⋅ θ , (8)

( )2 3b e ee K f= ω⋅ ⋅ θ − π , (9)

( )4 3c e ee K f= ω⋅ ⋅ θ − π , (10)

where θe is the electric angle of the motor, ω is the angular speed of the rotor, Ke is constant for electromotive voltages (V/(rad / sec)) and the reference function f(θe) for electric tension is alternatively trapezoidal with amplitude 1 (Fig. 3).

θe

θe

θe

π2π

3π

π−θ

32

ef

π−θ

34

ef

ia

ib

ic ic

ib

f(θe)

( )34π−θef

( )32π−θef

f(θe) ia

1

-1

Iref

-Iref

1

-1

Iref

-Iref

1

-1

Iref

-Iref

θe Hall A

θe Hall B

θe Hall C

Fig. 3. Electromotive voltages, phase currents and position transducer outputs Hall



3

Phase alternating currents are rectangular, with alternations during the 2π/3 centered on the electric voltages (Fig. 3).

Taking the origin of the phase as shown in Fig. 3, the function f(θe) defined on intervals has the next expression:

π

π∈θ

π−θ

π+−

ππ∈θ−

π

π∈θ

π−θ

π−

π∈θ

=θ

2,3

5for 3

561

35,for 1

,3

2for 3

261

32,0for 1

)(

ee

e

ee

e

ef (11)

To the voltage equations are added the equation of motion

sdm J B mdtω

= ⋅ + ⋅ω + (12)

where m is the electromagnetic torque developed by the motor, ms is the static torque, J is the total moment of inertia (motor plus load), and B is the friction coefficient.

Also, the electromagnetic torque developed by the motor has the expression (Kennedy and Eberhart, 1995):

( ) ( ) ( )[ ]3/43/23/ π−θ+π−θ+π−θ=

=ω

++=

ecebeae

ccbbaa

fififiK

ieieiem

(13) Torque is mainly influenced by the waveforms of the electromotive voltages induced in the stator of electric motors due to rotor motion. Ideally, electric tensions have trapezoidal waveforms and stator currents are rectangular (Fig. 3), resulting a constant torque. In practice, there are pulsations of torque due to design imperfections that lead to removal from electric tensions perfect form trapezoidal or due to PWM control method, switching and hysteresis. Taking into account of connection between electrical angle θe, mechanically angle θm, and the number of pole pair’s p, we have the next relation:

e mpθ = ⋅θ , (14)

and that

mddtθ

ω = , (15)

Operating equations (5) - (7), (12) and (15) can be written as the state equations, respectively:

( ) 1e ea sa a

s m s m s m

K fdi R i udt L L L L L L

θ= − − ω +

− − − (16)

( )2 3 1e eb sb b

s m s m s m


θ − π= − − ω +

− − − (17)

( )4 3 1e ec sc c

s m s m s m


θ − π= − − ω +

− − − (18)

scee

bee

aee

mJJ

BiJ

fK

iJ

fKiJfK

dtd

1)3/4(

)3/2()(

−ω−π−θ

+

+π−θ

+θ

=ω

(19)

ed pdtθ

= ω . (20)

In the form of a matrix, the equation of state is:

uBA ⋅+⋅= ξξ ; (21)

the vectors of inputs (u) and state variables ( ξ ) are:

[ ]smcubuau=u (22) [ ]ecibiai θω=ξ (23)

and the matrices A and B have the form:

( )

( )

( )

( ) ( ) ( )

0 0 0

2 30 0 0

4 30 0 0

2 3 4 30

0 0 0 0

e es

s m s m

e es

s m s m

e es

s m s m

e e e e e e

K fRL L L L

K fRL L L L

K fRL L L L

K f K f K f BJ J J J

p

θ − − − −

θ − π− −

− − = θ − π − − − −

θ θ − π θ − π −

A

(24)

1 0 0 0

10 0 0

10 0 0

10 0 0

0 0 0 0

s m

s m

s m

L L

L L

L L

J

− −

= −

−

B

(25)

3. THE GENERATION OF RESIDUE DURING THE MODEL-BASED DIAGNOSIS

When the system has the actuators affected by faults, this situation can be described by the relation:

++=++=

)()()()()()()()(tututxtytututxtx

d

dDDCBBA (26)

where ( )tx is the system’s state, ( )ty is the system’s output, ( )tu is the system’s command, A, B, C and D are constant matrices of appropriate dimensions and the vector ( )tud represent the fault vectors for the actuators. The transfer function type input-output representation for the system is the following:



4

)()()()()( susHsusHsy d+= (27)

where

DBAIC +−= −1)()( ssH (28)

The residue generator is a linear processor whose inputs consist in the input and output of the monitored system. This structure can be expressed mathematically so (Fig. 4):

[ ] )()()()()()(

)()()( sysQsusPsysu

sQsPsr +=

⋅= (29)

Fig. 4. The generation of the residue.

The matrixes ( )sP and ( )sQ are transfer matrixes built using linear, stabile systems. According to the definition, the residue is designed to become 0 in the case of fault absence and different from 0, in the presence of faults.

( ) ( ) 0 ifonly and if 0 == tutr d (30) For the residue generator ( )sr to be a fault indicator, the transfer matrixes ( )sP and ( )sQ must satisfy the relation:

0)()()( =+ sHsQsP (31)

When the system has the sensors affected by faults, this situation can be expressed using next structure (Fig. 5).

Let to consider a dynamical process of the form given by the linearised equations:

)()()( tutxtx cBA +=•

(32)

where nx ℜ∈ is the state vector, mu ℜ∈ is the known input vector, and A, B are constant matrices of appropriate dimensions. Similarly, the mathematical model of the process is:

)()()( tutxtx cmm BA +=•

(33)

The residual vector r(t) is generated by the equation:

)()()( txtxtr m−= (34)

( ) ( ) 0 ifonly and if 0 == tutr f (35)

Ideally, in absence of a fault, the residual should be zero, while when a fault is present it should be different from zero. So, a fault detection test will consist in check if the residual is zero or not.

4. EXPONENTIAL SMOOTHING METHOD

Single exponential smoothing is used for smoothing discrete time series. The efficiency of this algorithm can be attributed to its simplicity and to the capacity to adjust its responsiveness to changes in the process and its reasonable accuracy.

Let be an observed time series { }nxxxX ...21= . Formally, the simple exponential smoothing equation takes the form (Ostertagová, 2011):

ixixix~)1(1

~ αα −+=+ (36)

where ix is the actual, known series value at moment time i, ix~ is the forecast value of the variable X at time i,

1~

+ix is the forecast value at time i+1 and α is the smoothing constant.

H(s)

H(s)

P(s) Q(s)

ud(s)

u(s)

y(s)

r(s)

Process

Model

uf1 ufn

+

+

+ +

+

+ -

-

uc1 ucm

x1

xn

xm1

xmn

r1

rn

Fig. 5. Method for generates the residual vector for the sensor diagnosis.



5

Smoothing constant α is a selected number between zero and one, 0



6

Prasad, G., Sree Ramya, N., Prasad, P.V.N., and Tulasi Ram Das, G. (2012). Modelling and Simulation Analysis of the Brushless DC Motor by using MATLAB, Int. J. of Innovative Technology and Exploring Engineering, Vol. 1, Issue 5, Oct. 2012.

Iancu E. and Vinatoru M. (2005). A fault detection and isolation system using neural networks, Proceedings of the International Conference on Control Systems and Computer Science CSCS 15, Bucuresti, vol. 1, pp. 422-427.

Kennedy, J. and Eberhart, R.C. (1995). Particle swarm optimization. Proc. of IEEE Int. Conference on Neural Networks, Piscataway, NJ. pp. 1942-1948.

Nguang S.K., Shi P., Ding S. (2005). Fault detection filter for uncertain fuzzy systems: an LMI approach, IFAC Congress, Praha, CD proceedings.

Ostertagová, E. and Ostertag O. (2011). The simple exponential smoothing model, Proceedings of the 4th International Conference on Modelling of Mechanical and Mechatronic Systems, Technical University of Košice, Slovak Republic, , pp. 380-384.

Patton R.J. and Chen J. (1994). A review of parity space approaches to fault diagnosis applicable to aerospace systems, Journal of Guidance, Control and Dynamics, vol. 17, no. 2, pp. 278-285.


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7

Fault Detection in Educational Kit Festo

Camelia Maican*, Gabriela Cănureci**

* Automation and Electronics Department, University of Craiova, Romania (e-mail: [email protected]).

** Research and Development, Engineering and Manufacturing for Automation Equipment and Systems SC IPA SA CIFATT Craiova,

Romania (e-mail: [email protected]).

Abstract: In this paper is study the faults detection and localization, in educational kit Festo, using residual methods. The level control system and the faults detection structure were developed under Matlab Simulink. This faults detection structure allows us to detect two faults that can occur in the plant, separately and simultaneously. The proposed method was theoretically developed and experimentally verified on the plant model.

Keywords: control; fault detection and localization; level; residue.

1. INTRODUCTION The plant Festo contains individual modules which can be combined in different ways and allows the level, temperature and flow control. Based on equivalence relations of flowing phenomena, one can determine the equivalence of the level controlling system within Festo plant with the level controlling system of the drums in the steam boilers within the thermal power groups (Canureci et al. 2012). Within both plants, there can be malfunctions like: - blocking the adjustment flap of the boiler feed water flow, or a changing transfer factor, which can be simulated on the Festo plant by modifying the flowing section of the tap on the circulation pump discharge. - pipe breaking or stopping of a power pump on the water flow of the boiler, which lowers the boiler feed water flow according to the value generated by the controller, fact that can be simulated on the plant by activating the tap R3, which partially controls the flow Fp directly to the second tank. This way, the structures of detection and identification of the malfunctions that may appear on the given plant can be applied analogically to the steam boiler level control system (Iancu et al. 2003 and Vinatoru 2001).

2. THE MATHEMATICAL MODELS The hydraulic system diagram of the educational kit FESTO is shown in Fig.1 (Canureci et al. 2012). The plant consists of two parallelepipedic transparent plastic water tanks assembled on an aluminium platform with supporting holders. The tanks are placed one in upper position the other in lower position. A water pump (P) ran by a driving motor (DM) ensures water flow from the lower tank to the upper tank through a system of pipelines, bends and connecting pipes. At the exit of the pump (P), on the pipe there is a vertically assembled hydraulic diode of connection which

prevents water leaks from the upper tank to the lower tank when the pump discharge pressure gets below the hydrostatic pressure which corresponds to the liquid level of the upper tank. A pipe system combined with two taps R1 and R2 allow conducting the water discharged by the pump either towards the upper tank or to the lower tank. Water flow from the upper to the lower tank is done naturally through a pipe system on which there is a tap R3 which allows changes in the flowing section or obstructions of the pipe (Vinatoru et al. 2008). On the upper tank there is mounted a level transducer with ultrasounds which determines an electric signal at the exit 4-20 mA DC for a fluctuation of the liquid level within the field 0-500 mm (Vinatoru et al. 2007). The liquid level control is achieved by changing the water pump flow (F3=Fp), using the speed command of the driving motor (DM).

Fig. 1 The hydraulic system diagram

L1

L2

DM UC

Tank 1

Tank2

R1

R2

F3

S2 R3 F2

P

Fp


____________________________________________________________________________________________________________

8

This is achieved by varying the supply voltage of the pump using the pump regulator, which is driven with a voltage signal in the range of 2-10 V DC.

The state variables are: x1 = L1 the level of the tank1 x2 = L2 the level of the tank2

The input variables are: F3 = Fp = U = kpUc the command to adjust the

level in the tank 1

121212 LCSgLSCF == ρ the evacuation flow from the tank 2.

The mathematical model of the system is determined using the mass balance equations (Vinatoru et al. 2008):

1

12

1

1221 ),()(A

LSCUkA

LSFUFdt

dL Cpcp −=−

= (1)

2

12

2

31222 )(),(A

UkLSCA

UFLSFdt

dL CpC −=−

= (2)

where C is a coefficient depending on the viscosity of the fluid loss and sectional shape of the flowing, S2 sectional area of the valve transitions between the two tanks, A1 and A2 cross-sectional area of the Tank1, respectively Tank2.

In canonical form the mathematical model is:

C

P UA

KLSAC

dtdL

112

1

1 +−= (3)

2

212

2

2 UAKLS

AC

dtdL P−=

(4)

In steady state:

2010200 FLSCUk Pp == (5)

By linearizing of the equations (3) and (4) around the steady state values, resulting linear form of the mathematical model (6) and (7), which contain the section variation ΔS2 which can occur only under fault conditions. Under normal conditions ΔS2 = 0.

2

1

10

11

101

201 2

SALC

UAk

xLA

CSx C

p ∆−∆+∆−=∆ (6)

2

1

10

11

101

202 2

SALC

UAk

xLA

CSx C

p ∆+∆−∆=∆ (7)

where: Δx1 = ΔL1 = L1 - L10

Δx2 = ΔL2 = L2 - L20,

x10 = L10 and x20 = L20

Applying Laplace transform in zero initial conditions result:

∆−∆

+=∆ )(2)(

21

1)( 220

10

20

101 sSS

xsUF

kxTs

sx Cp (8)

∆+∆−

+=∆ )(2)(

21

)( 220

10

20

10212 sSS

xsUF

kxTs

AAsx Cp (9)

where: 20

1012F

xAT =

and A1=A2=1,75∙1,9=3,325dm2

L10=L20=2dm

F10=F20=4,7dm3

3. THE FAULT DETECTION AND LOCALIZATION We will analyze the following faults which may occur in the FESTO system:

• The pump is not running completely the command received or an additional pressure loss occurs on the above valve, which is mounted after the pump, and this makes the pump flow to be influenced by fault, therefore be no longer proportional to the control voltage:

( )dCnpp UUkF ∆+∆=∆ (10) where ΔUCn is pump control signal and ΔUd the equivalent in the control signal of the actuator's fault.

• Modification of the flow section S2 due either to variation of the flow regime, by plugging the crossing section or additional breakings of pipeline, which is equivalent to an increase of S2; this makes as the exhaust flow from the Tank1 in Tank 2 to be of the form:

( )dn SSLCF 22102 ∆+∆=∆ (11) Where ΔS2n is the value of the section under normal functioning and ΔS2d is equivalent value of the section, caused by the fault.

In these circumstances, by introducing the faults specified in equations (6) and (7), replacing

ΔUC = ΔUCn + ΔUd

ΔS2 = ΔS2n + ΔS2d

we obtain:

( )

( )dn

dCp

SSALC

UUAk

xLA

CSx

221

10

11

101

201 2

∆+∆−

−∆+∆+∆−=∆ (12)


____________________________________________________________________________________________________________

9

( )

( )dn

dCp

SSALC

UUAk

xLA

CSx

221

10

11

101

202 2

∆+∆+

+∆+∆−∆=∆ (13)

Applying Laplace in equations (12) and (13) in zero initial conditions and using the transformations and the notations from equations (8) and (9) it follows the operational form of the real process equations in which faults were included.

Following the same steps to equations (12) and (13) as the relations (6) and (7) it results:

∆−

∆−

−∆+∆

+=

20

210

20

210

20

10

20

10

1 )(2

)(2

)(2

)(2

11)(

SsS

xS

sSx

sUF

kxsU

Fkx

Tssx

dn

dp

Cnp

r(14)

∆+

∆+

+∆−∆−

+=

20

210

20

210

20

1

20

10

212 )(

2)(

2

)(2

)(2

1/

)(

SsS

xS

sSx

sUF

kxsU

Fkx

TsAA

sxdn

dpr

Cnp

r(15)

where: TLA

CS 12 101

20 =

From equations (8) and (9) it follows the process model, considering ΔS2d = 0:

∆−∆

+=

20

210

20

101

)(2)(

21

1)(S

sSxsU

Fkx

Tssx nCn

pm (16)

∆+∆−

+=

20

210

20

102

)(2)(2

11)(

SsSxsU

Fkx

Tssx nCn

pm

(17)

We define the residues r1(s) and r2(s) (Gertler et al. 2002 and Korbicz et al. 2004):

( ) ( ) ( )sxsxsr mr 111 −=

( ) ( ) ( )sxsxsr mr 222 −=

( ) ( ) ( ) ( )( )

20

210

20

101 21

121

1S

sSx

TssU

Fkx

Tssr dd

p ∆+

−+

= (18)

( ) ( ) ( )( )sSsSx

TssU

Fkx

Tssr dd

p

20

210

20

102 21

121

1 ∆+

+∆+

−= (19)

From the equations (18) and (19) it can be seen that the residues r1 and r2 are both of influenced by the faults ΔS2d and ΔUd (Table.1).

Table 1. Influence of faults to the residues Residues/faults ΔUd ΔS2d r1 1 1 r2 1 1

In this case, the fault matrix is:

( ) ( )

( )

++−

+−

+=

12

12

12

12

10

20

10

10

20

10

Tsx

TsFkx

Tsx

TsFkx

sGp

p

D (20)

where: 2,17,4

4,1222

20

10 =⋅⋅

=⋅⋅

Fkx p [dm/V]

8,22

20

101 ==F

xAT [min]

( )

++−

+−

+=

18,24

18,22,1

18,24

18,22,1

ss

sssGD (21)

From the equation (21) it can observe that the fault matrix is non-invertible, because detGD(s) = 0.

Accordingly, we use only the residue r1 with which we can determine the fault ΔUd or the fault ΔS2d.

Because: ( )18,2

2,1+

=s

sGD

Result: ( )2,1

18,21 +=− ssGD

Because GD-1(s) cannot be physically realized, for implementation, is introduces a pole, with a small time constant, and in this case is chosen:

( ))1(2,1

18,2*1++

=−sssGD

(22)

To detect and locate the faults was achieved the simulation scheme shown in Fig.2, with the following blocks:

• In the upper part is simulated the Festo system, with the level control structure of the Tank 2 commanding of the pump flow, which feed the Tank 1. In the system are simulated the fault ΔUd representing the variation of pump flow and the fault ΔS2d representing an additional flow variation drain from Tank 1.

• In the lower part is the Festo model, functioning in normal condition, to which the same command provided by PI controller is applied.


____________________________________________________________________________________________________________

10

Fig. 2 The fault detection structure Also in the lower part were generated the residues:

r1=L1r-L1m

r2=L2r-L2m (23)

r3=r1+r2

Simulation results are presented in the next figures.

In Fig 3and Fig. 4 are represented the level for the real plant, respective the residues in normal condition.

0 10 20 30 40 50 60 70 80 90 1001.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

Fig. 3 The level L1r and L2r in normal condition to the change of the prescribed size

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fig. 4 The residues r1 and r2 in normal condition to the change of the prescribed size

0 10 20 30 40 50 60 70 80 90 1001.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

Fig. 5 The level L1r and L2r for Δ Ud=0.1

0 10 20 30 40 50 60 70 80 90 100-1.5

-1

-0.5

0

0.5

1

1.5

Fig. 6 The residues r1 and r2 for Δ Ud=0.1 In Fig 5and Fig. 6 are represented the level for the real plant, respective the residues in fault condition, were the residue is Δ Ud=0.1, applied at time t = 40 s.

In Fig 7and Fig. 8 are represented the level for the real plant, respective the residues in fault condition, when ΔUd=0.5, applied at time t = 40 s.

In both case the residues are indicate the fault.

r1 r2

Residues

Time [s]

L2r

L1r

Level [dm]

Time [s]

r1

Residues

Time [s]

r2

ΔUd

L2m L1m

L1r

ΔSd

ΔUd

•

L2r L1r*

• •

• PI Actuator

Real Plant

r2

r1

Model

Plant

r3

Actuator

ΔUd

L2r

L1r

Level [dm]

Time [s]

ΔUd


____________________________________________________________________________________________________________

11

0 10 20 30 40 50 60 70 80 90 1001.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

Fig. 7 The level L1r and L2r for Δ Ud=0.5

0 10 20 30 40 50 60 70 80 90 100-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Fig. 8 The residues r1 and r2 for Δ Ud=0.5

0 10 20 30 40 50 60 70 80 90 1001.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

Fig. 9 The level L1r and L2r for Δ S2d=0.05 Fig 9 and Fig. 10 represented the level for the real plant, respective the residues in fault condition, when ΔS2d=0.05, applied at time t = 50 s. The new residue r3 indicate this fault and the fault value. Fig 11 and Fig. 12 represented the level for the real plant, respective the residues in fault condition, when Δ S2d=0.1, applied at time t = 70 s.

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 10 The residues r1, r2 and r3 for Δ S2d=0.05

0 10 20 30 40 50 60 70 80 90 1001.5

2

2.5

3

3.5

4

Fig. 11 The level L1r and L2r for Δ S2d=0.1

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 12 The residues r1, r2 and r3 for Δ S2d=0.1 In Fig. 13 is presented the response of the real system, using the variables, L2r- level in the tank2, which is changed as a result of the command given by the controller corresponding to the control loop of L2r, so as to ensure the flow between L1 and L2 equal to the pump flow.

r1

Residues

Time [s]

r2 ΔUd

r1

Residues

Time [s]

r2 ΔS2d

r3

ΔS2d

L2r

L1r

Level [dm]

Time [s]

r1

Residues

Time [s]

r2

ΔS2d

r3

ΔS2d

L1r

Level [dm]

Time [s]

L2r

L2r

L1r

Level [dm]

Time [s]


____________________________________________________________________________________________________________

12

We notice that both of defects do not produce variations to the controlled value L2r, them being compensated by the control system. To the L1r variable can be determined only the fault ΔS2d, applied at time t = 70 s, by its variation. So just based on variations of L1r and L2r cannot detect the defects, at most we can interpret the variation of L1r if there is prior knowledge about the influence of ΔS2d upon L1r.

Instead the variations of the residues r1, r2 and r3 defined above, clearly highlights these defects. However, the defect ΔUd strongly influences both the residues r1 and r2, but ΔS2d has influence on residue r3.

So we cannot locate both faults using residues r1 and r2. In this case, the residue r3 is defined as:

( ) ( ) ( )( ) ( ) ( )[ ]( )( ) ( ) ( )sSsZsrsr

sWsWsrsr

sWsr d22

11211

2

13 ∆=

=

=

From the transfer matrix GD(s) in the steady state resulting W11 = W12 = 1 and r3(s) is influenced only by the fault ΔS2d. This is graphically represented in Fig.14.

0 10 20 30 40 50 60 70 80 90 1001.8

2

2.2

2.4

2.6

2.8

3

3.2

Fig. 13 The level L1r and L2r for ΔUd=0.1 and Δ S2d=0.05

0 10 20 30 40 50 60 70 80 90 100-1

-0.5

0

0.5

1

1.5

Fig. 14 The residues r1, r2 and r3 for ΔUd=0.1 and ΔS2d=0.05

4. CONCLUSIONS

In this paper a method for faults detection and localization using residual vectors was presented; the proposed method was theoretically developed and experimentally verified. It allowed detection and localization of two faults created in a real process. A series of case studies was realized regarding the possibilities for detection of the faults using residuals for the system.

With the fault detection and localization scheme presented in this paper it can be detected the fault ΔUd by the residue r1 and the fault ΔS2d by the residue r3.

The advantage of this detection structure is that residue r3 show us the fault value.

ACKNOWLEDGMENT

This work was supported by the strategic grant POSDRU/159/1.5/S/133255, Project ID 133255 (2014), co-financed by the European Social Fund within the Sectorial Operational Program Human Resources Development 2007 - 2013.

REFERENCES

Canureci, G., Vinatoru, M., and Maican C. (2012). Fault detection and localization in dynamical systems, Ed. SITECH, Craiova

Gertler, J., Staroswiecki M., and Shen, M. (2002). Direct Design of Structured Residuals for Fault Diagnosis in Linear Systems, American Control Conference, Anchorage, Alaska.

Iancu E. and Vinatoru M. (2003). Analytical method for fault detection and isolation in dynamic systems study case, Ed. Universitaria Craiova.

Korbicz, J., Koscielny J. M., Kowalczuk Z., and Cholewa W. (2004). Fault diagnosis, Ed. Springer.

Vinatoru, M. (2001). Automatic control of the industrial process, Ed. Universitaria, Craiova.

Vinatoru, M., Canureci, G., Maican C., and Iancu, E. (2007). Level and Temperature Control Study Using Festo Kit in a Laboratory Stand, Proceedings of the 3rd WSEAS/IASME International Conference on Dynamical Systems and Control (CONTROL’07), Arcachon, Franta, pg. 247-252.

Vinatoru, M., Iancu, E., Canureci G., and Maican, C. (2008). Automatic control of the industrial process - Design wizard and laboratory, vol. 2, Ed. Universitaria Craiova.

ΔS2d

ΔUc

L2r

L1r

Level [dm]

Time [s]

ΔUc

r1

Residues

Time [s]

r2

ΔS2d

r3



13

Compact Dynamic Model of the Brushless DC motor

Sergiu Ivanov*, Dan Selişteanu**, Virginia Ivanov*, Dorin Şendrescu**

* Faculty of Electrical Engineering, University of Craiova, Romania

(e-mail: [email protected], [email protected]). ** Faculty of Automation, Computer and Electronics, University of Craiova, Romania

(e-mail:{dansel, dorins}@automation.ucv.ro)

Abstract: Both the researchers and manufacturers are interested by the light urban electric vehicles. More ways are investigated for the used motors, the most important concern being the compactness of the solution. Thanks to the high power density, the most preferred motors are the permanent magnet synchronous motors (PMSM) and brushless DC motors (BLDC). These are preferred also thanks to their high efficiency and low maintenance cost. For both, the main manufacturing technology and associated power electronics are quite similar. The differences occur in the controlling technology, for BLDC being simpler and more advantageous. As integration technology, the direct drive in-wheel technology improves the safety, efficiency, weight, controllability and finally the costs. The development of the control strategies implies the need for a simple and compact model of the BLDC. The paper deals with an efficient dynamic model of this type of motor. It will be applied for the most used control strategy, the preset currents respectively.

Keywords: BLDC motor model, S-function, in-wheel motor, control.

1. INTRODUCTION The brushless DC (BLDC) motors become in the last years quite interesting for different industrial and home applications (servo drives, peripherals, small vehicles), thanks to a series of advantages related to the small volume, high power density (light-weight), efficiency (Yedamale, 2003).

More communications report the use of such type of motor for traction purposes (Tashakori et al., 2011). The direct drive in-wheel technology is preferred more and more for low scale traction applications. The inclusion of separated motors in each wheel eliminates the need of a central drive and the corresponding complicated, imprecise and heavy mechanisms necessary for distribute the torque to the wheels (Tashakori et al., 2010).

The BLDC motor is quite similar as construction with the permanent magnets synchronous one. Consequently, the specified advantages are common. The difference occurs in the control technique. For the BLDC is much simpler, as it needs only the identification of the one of the six 60º sectors where the rotor is situated. Contrary, the PMSM needs an absolute, precise position encoder.

Several publications (Rambabu, 2007; Prasad et al., 2012; Fedák et al., 2012, Singh et al., 2013) deeply describe the mathematical model of the BLDC. The practical implementation depends on the researchers experience and technical options.

Concerning the command, the natural or phase variable model offers many advantages thanks to the trapezoidal

back EMF, which avoids the well-known Park transformation necessary when sinusoidal variation of the motor inductances with rotor angle occur, as is the case of the PMSM.

The paper presents an original and very compact implementation of the BLDC model by using the S-function facility of Simulink. The most used control strategy is implemented by using the developed motor model, confirming its viability.

2. BLDC MOTOR MODEL By considering several hypothesis, the phase variable model is implemented: the stator phase resistances are constant (is neglected the temperature influence), all the inductances are constant (magnetic saturation is not present), hysterezis and eddy current losses are not considered and all the inverter switches are ideal.

With the above assumptions, the dynamic model of the BLDC motor is described by the voltage equations on the three phases:

( ) aa s a s m adi

u R i L L edt

= + − + , (1)

( ) bb s b s m bdi

u R i L L edt

= + − + , (2)

( ) cc s c s m cdi

u R i L L edt

= + − + . (3)

In order to obtain the dynamic model in the state variables form, it is advantageous to write (1-3) in matrix form



14

[ ] [ ][ ] [ ] [ ] [ ]s mdu R i L L i edt

= + − + , (4)

where

[ ] [ ]Ta b cu u u u= is the input voltages vector;

[ ]0 0

0 00 0

s

s

s

RR R

R

=

is the stator resistances diag. matrix;

[ ] [ ]Ta b ci i i i= is the phases currents vector;

[ ]0 0

0 00 0

s m

s m s m

s m

L LL L L L

L L

− − = − −

is the stator

inductances diagonal. matrix; [ ] [ ]Ta b ce e e e= is the back emf vector.

With this matrix form, the state space model is simply obtained:

[ ] [ ] [ ] [ ][ ] [ ]( )1s mdi i L L u R i edt− = = − − − (5)

Taking into account the diagonal form of the inductances matrix, its inverse has as elements, the inverse of each element.

The back emf vector [e] has as elements:

( )a e ee K f= ω⋅ ⋅ θ , (6) ( )2 3b e ee K f= ω⋅ ⋅ θ − π , (7) ( )4 3c e ee K f= ω⋅ ⋅ θ − π , (8) where θe is the electric angle of the rotor, ω is the angular speed of the rotor, Ke back emf constant [V/(rad/sec)] and the reference function f(θe) of the back emf is alternative, with trapezoidal shape and amplitude equal to 1. In Fig. 1 are plotted the “a” phase waveforms of the reference function fa(θe), preset current ia* and the Hall signal.

Fig. 1. The waveforms for the phase a. For the other two phases, all the signal are delayed by 2π/3 and 4π/3 respectively:

( ) 23b e a e

f f π θ = θ − ,

* * 23b a e

i i π = θ − ,

2Hall Hall3b a eπ = θ −

,

( ) 43c e a e

f f π θ = θ − ,

* * 43c a e

i i π = θ − ,

4Hall Hall3c a eπ = θ −

. The model described by (5) is completed with the torque expression

( ) ( ) ( ) =

a a b b c c

e a a e b b e c c e

e i e i e im

K i f i f i f

+ += =

ω⋅ ⋅ θ + ⋅ θ + ⋅ θ . (9)

and with the movement equation,

sdm J B mdtω

= ⋅ + ⋅ω +. (10)

where m and ms are the electromagnetic torque and the static one respectively, J is the total inertia and B is the viscous frictions coefficient.

From (10) it results the fourth state equation:

( )1 s

d m m Bdt Jω

ω = = − − ⋅ω. (11)

The state variables will be the phase currents and the speed.

The link between the electric angle θe and the mechanical one is done by the number of pairs of poles

e mpθ = ⋅θ . (12) The mathematical model described by (5-9, 11-12) was implemented by using the S-function facility of Simulink. Fig. 2 depicts the result.

Fig. 2: The Simulink model of the BLDC with S-function. The highlighted areas correspond to the computation of the electromagnetic torque m, by using (9) and to the integration of the mechanical state equation (11).

All the dynamic model described by (5) is written in an m-file (Fig. 3) which is called with different flags by the S-function block during the simulation. In the zone 1, the inputs (phase voltages) are updated. In the zone 2, the state derivatives are computed with (5). In zone 3 the outputs (phase currents) are computed.

The block tem from Fig. 2 generates, based on the electric angle reduced to the 0-2π range, the reference functions of the back emf (6-8) shown in Fig. 4, used both for integration of the dynamic model (5) and for electromagnetic torque computation (9).

θe π2

π/ 6π

*ai ( )a ef θ

f(θe) ia 1

-1

I*

-I*

θe Halla

5 / 6π

7 / 6π 11 / 6π

m

mec



15

Fig. 3: The S-function corresponding to the BLDC motor.

Fig. 4: The reference back emf functions fa, fb, fc. As can be seen, the Simulink model is quite intuitive and compact. The values of all the parameters are transferred by the dialog box (Fig. 5) of the grouped model shown in Fig. 2.

Fig. 5: Dialog box of the BLDC motor model

3. COMMAND AND SIMULATIONS Basically, the command grants the classical 120º square currents, in phase with the back emf, in order to maximize the developed torque.

The experienced method is based on the preset current modulation, which determines that the inverter is a current source.

The principle of this type of command is depicted in Fig. 6.

1

2

3



16

Fig. 6: Principle diagram for the preset currents command Based on the information delivered by Hall sensors on the motor shaft, the block “fi Generator” generates, the 120º square waveforms of the currents in p.u. The amplitude of the currents is obtained as result of the PI speed controller which multiplies the unitary waveforms in order to obtain the preset stator currents. These are compared then with the real ones by using three hysterezis comparators, like in the well-known bang-bang modulation. The result pulses are applied to the six switches of the inverter bridge.

Fig. 7 plots few results of the simulation: step start at no load (0.5 Nm) and 10 rad/sec, followed by a load application (1.5 Nm) at 0.14 s and an acceleration to 15 rad/sec with the applied load.

Fig. 7: Results for preset currents command We notice the very good dynamic behaviour, the command and waveforms accuracy. Only the motor inertia was considered.

The command method has the advantage to be quite simple to be implemented, as low cost analogue hysterezis controllers can be used. Also, the method is not

sensitive to the variations of the motors parameters, especially stator resistance and to the variations of the supplying voltage. The main disadvantages are related to the variable switching frequency and application size, as there are necessary fast switching elements, available only at low to medium power.

4. CONCLUSIONS The paper presents a compact and simple implementation of the BLDC motor dynamic model, by using the S-function facility of Simulink. One type of command is implemented in order to test the model viability. The advantages and disadvantages are emphasized and possible applications highlighted. The future activity will be focused on the development of different commands types.

ACKNOWLEDGMENT

This work was partially supported by the grant number P09004/1137/31.03.2014, cod SMIS 50140, entitled: Industrial research and experimental development vehicles driven by brushless electrical motors supplied by lithium-ion accumulators for people transport - GENTLE ELECTRIC.

REFERENCES Fedák, V., Balogh, T., and P. Záskalický. (2012). Dynamic

Simulation of Electrical Machines and Drive Systems Using MATLAB GUI- A Fundamental Tool for Scientific Computing and Engineering Applications - Volume 1, Prof. Vasilios Katsikis (Ed.), ISBN: 978-953-51-0750-7, InTech.

Prasad, G., Sree Ramya, N., Prasad, P.V.N., and Tulasi Ram Das G. (2012). Modelling and Simulation Analysis of the Brushless DC Motor by using MATLAB, International Journal of Innovative Technology and Exploring Engineering, Vol. 1, Issue 5.

Rambabu, S. (2007). Modeling and control of a brushless DC motor, PhD Thesis.

Singh, C.P., Kulkarni, S.S., Rana, S.C., and Kapil D. (2013). State-Space Based Simulink Modeling of BLDC Motor and its Speed Control using Fuzzy PID Controller. International Journal of Advances in Engineering Science and Technology, Vol. 2 , No. 3, pp. 359-369.

Tashakori, A., Ektesabi, M., and Hosseinzadeh, N. (2010). Characteristic of suitable drive train for electric vehicle. In Proceedings of the 3rd International Conference on Power Electronic and Intelligent Transportation System (PEITS), 20-21 Nov. 2010, Shenzhen, China.

Tashakori, A., Ektesabi, M., and Hosseinzadeh, N. (2011). Modeling of BLDC Motor with Ideal Back-EMF for Automotive Applications. In Proceedings of the World Congress on Engineering, WCE 2011, July 6-8, 2011, London, U.K.

Yedamale, P. (3003). Brushless DC (BLDC) Motor Fundementals. AN885, 2003. Michrochip Technology Inc.



17

Adaptive and Predictive Control Algorithms for a Microalgae Process

Emil Petre

Department of Automatic Control and Electronics, University of Craiova, Craiova, Romania (e-mail: epetre@ automation.ucv.ro)

Abstract: This paper deals with the design and the analysis of two control structures for microalgae culture process to regulate the substrate concentration at a chosen setpoint. The control strategies are developed under the realistic assumptions that the reaction rates are time varying and incompletely known and the influent substrate concentration as well as the light intensity are strongly time-varying. The first control structure is an adaptive control algorithm. This controller is designed by coupling a linearizing controller with a parameter estimator used for estimation of unknown kinetics. The predictive control structure is based on classical nonlinear model predictive control law under model parameter uncertainties implying solving a nonlinear least squares optimization problem for setpoint trajectory tracking. The proposed approaches are validated in simulation and numerical results are given to illustrate its efficiency for setpoint tracking in the presence of parameters uncertainties. Keywords: Bioprocesses, Microalgae, Droop model, Photobioreactor, Adaptive control, Predictive control, Nonlinear least squares optimization.

1. INTRODUCTION It is well known that water is an essential element of life being an important resource both for industrial applications and domestic usage. Therefore in last time numerous environmental laws and directives have arisen in order to decrease the pollution related to the industrial and urban effluents. This situation has led to an increase in the use of wastewater biological treatment processes. For example, the anaerobic digestion is very useful since it produces valuable energy (methane) besides removing the organic pollution from the liquid influent (Angulo et al., 2007). Nevertheless, its main drawback is the production of carbon dioxide (CO2) and its easy destabilization (Angulo et al., 2007; Bastin and Dochain, 1990; Bernard, 2004). Therefore the researchers have been searched solutions to improve the efficiency in the pollution reduction and for CO2 mitigation (Bernard, 2011; Benattia et al., 2014a, b; Ifrim et al., 2013). A recently used solution consist in the growth of some microalgae populations (in fact, autotrophic microalgae and cyanobacteria (Bernard, 2011) that by using light as source of energy are able to assimilate inorganic forms of carbon (CO2, −3HCO ) and to convert them into requisite organic substances intended to many industrial applications: food, pharmacology, chemistry, generating at the same time oxygen (O2) (Bernard, 2011; Benattia et al., 2014a, b; Ifrim et al., 2013). Microalgal biofuel production systems could also contribute to mitigate industrial CO2 emitted from power plants, cement plants, etc. In the same spirit, microalgae could be used to consume inorganic nitrogen and phosphorus in urban or industrial effluents, and thus limit expensive wastewater post-treatment (Bernard, 2011). However, microalgae have been so far only marginally used for biotechnological applications as: vitamins, proteins, cosmetics, and health foods. But in perspective of large scale microalgal cultivation, the modelling and

control of such processes remains a key issue for the enhancement of stability and process efficiency since microalgae have some specificities compared to microorganisms such as bacteria or yeasts (Bernard, 2011). A difficulty for the design of high-performance control techniques of such living processes lies in the fact that, in many cases, the models contain kinetic parameters and/or yield coefficients that are highly uncertain and time varying (Bastin and Dochain, 1990; Batstone et al., 2002; Bernard, 2011; Dochain and Vanrolleghem, 2001; Mairet et al., 2011). Therefore most of them must to be estimated (Bastin and Dochain, 1990; Dochain and Vanrolleghem, 2001). To control these processes, several control strategies were developed such as linearizing feedback (Angulo et al., 2007; Bastin and Dochain, 1990; Dochain, 2008; Neria-González et al., 2009; Petre et al., 2013; Petre and Selişteanu, 2013; Tebbani et al., 2014a), adaptive and robust-adaptive approach (Bastin and Dochain, 1990; Neria-González et al., 2009; Petre et al., 2013; Petre and Selişteanu, 2013), predictive an optimal control (Bernard, 2011; Benattia et al., 2014a, b; Logist et al., 2011; Tebbani et al., 2014, 2015), sliding mode (Selişteanu et al., 2007), neural techniques (Hayakawa et al., 2008; Petre et al., 2010), and so on. Some of these approaches imposed the use of the so-called “software sensors”, used not only for the estimation of concentrations of components but also for the estimation of kinetic parameters or even reaction rates (Bastin and Dochain, 1990; Dochain and Vanrolleghem, 2001; Neria-González et al., 2009; Petre et al., 2013; Petre and Selişteanu, 2013). The aim of this paper is to develop some adaptive and predictive control structures, able to deal with the model uncertainties, for a microalgae culture process that is carried out in a continuous perfectly mixed bioreactor. The control algorithms are developed under the realistic assumptions that the reaction rates are time varying and incompletely known and the influent substrate



18

concentration as well as the light intensity are strongly time-varying. The adaptive control structure is achieved by combining a linearizing control law with a parameter estimator which plays the role of a software sensor for on-line estimation of unknown bioprocess kinetic rates. The predictive control structure is based on Nonlinear Model Predictive Control (NMPC) strategy (Camacho and Bordons, 2004). The main advantage of NMPC law is that it allows the current control input to be optimized, while taking into account the future system behaviour. This is achieved by optimizing the control profile over a finite time horizon, but applying only the current control input (Benattia et al., 2014a, b). The behaviour and performance of the proposed control algorithms are illustrated by numerical simulations applied in the case of a continuous microalgae bioprocess for which the bacterial growth rate is strongly nonlinear and time varying, the uptake rate is completely unknown, and the influent substrate concentration as well as the light intensity are strongly time-varying.

2. PROCESS DESCRIPTION AND MODELLING The well-known model of microalgal growth is the Droop model. This model takes into account the specificity of microalgae in comparison to other microorganisms that is they use light to grow and inorganic substrate uptake and growth are decoupled thanks to an intracellular storage of nutrients (Bernard, 2011; Benattia et al., 2014a, b). The Droop model involves three state variables: the biomass concentration X, in μm3 L−1, the limiting substrate (dissolved inorganic nitrogen (nitrate or ammonium)) concentration S, in μmol L−1, and the internal quota Q, in μmol μm−3, defined as the quantity of substrate per unit of biomass (Bernard, 2011; Benattia et al., 2014a, b). The considered dynamic model assumes that the process takes place in a perfectly mixed continuous photobioreactor (the influent flow rate equals the effluent flow rate, leading to a constant effective volume), without any additional biomass in the feed, and neglecting the effect of gas exchanges. Then the differential equations which describe this process (resulting from mass balances) are given by (Benattia et al., 2014a, b):

))(()())(()(

)())(),(())(()(

)()())(),(()(

tSSDtXtStS

tQtItQtStQ

tXDtXtItQtX

in −+ρ=

µ−ρ=

−µ=

(1)

where D is the dilution rate (d-1, d: day) and Sin the influent substrate (inorganic nitrogen) concentration (µmol L-1). The specific uptake rate )(Sρ is represented by a Monod model (Benattia et al., 2014a, b):

)()()(

tSKtSS

sm +

ρ=ρ , (2)

where mρ and Ks represent respectively the maximal specific uptake rate and the substrate half saturation constant. The specific growth rate ),( IQµ is modelled as (Benattia et al., 2014a, b):

))(()(

1),( tItQ

KIQ I

Q µ

−µ=µ , (3)

where µ is the theoretical specific maximal growth rate, i.e. the growth rate at hypothetical infinite quota (Bernard, 2011), KQ represents the minimal cell quota allowing growth, and )(IIµ denotes the light effect. This is modelled by a Haldane model which describes the photo-inhibition phenomenon as (Benattia et al., 2014a, b):

iIsI

I KIKIII

/)( 2++

=µ , (4)

where I is the light intensity (in µE m-2 s-1) and KsI and KiI are light saturation and inhibition constants respectively. The optimal light intensity that maximises the function

)(IIµ is given by iIsIopt KKI = (Benattia et al., 2014a). In the sequel, we will consider that the light intensity value is not set at this optimal value Iopt ; it is assumed a strongly time varying function (that should simulate some cloudy or rainy days and nights). The meaning and the values parameters in the Droop model are given in the Table 1 (Benattia et al., 2014b).

Table 1. Parameter values in the Droop model. Parameter Value and unit Meaning

mρ 9.3 μmol μm−3 d-1 Maximal specific uptake rate

sK 0.105 μmol L−1 Substrate half saturation constant

µ 2 d-1 Theoretical specific maximal growth

rate QK 1.8 μmol μm

−3 Minimal cell quota

sIK 150 μE m−2 s-1 Light saturation constant

iIK 2000 μE m−2 s-1 Light inhibition constant

inS 100 μmol L−1 Microalgae substrate concentration

3. CONTROL STRATEGIES

For the microalgae process described by the dynamical model (1)-(4) we consider the control problem of the internal substrate concentration at a chosen setpoint by using as control input the dilution rate D, the aim being also the obtaining of a great quantity of biomass. The control problem will be resolved under some realistic conditions specified in the previous section. 3.1. Exact linearizing control law Firstly, we consider the case when the whole bioprocess is completely known (the ideal case). This means that the model (1)-(4) is completely known (i.e. all the specific rates are assumed completely known, and all the state variables and the inflow rate are available by on-line measurements). Since in model (1) the dynamics of S has the relative degree equal to 1, then the following exact linearizing control law:

( ) ( )XSSSSSStD in )()(/1)( ** ρ+−λ+⋅−= , (4) where S* denotes the desired set point assures a stable behaviour of closed-loop system described by the following first order linear stable differential equation:



19

0)()( ** =−λ+− SSSS , 0>λ . (5)

where λ is a design parameter. The control law (4) leads to a linear dynamics of the tracking error yye −= * described by ee λ−= , which for 0>λ has an exponential stable point at 0=e . This control law will be used in order to design the adaptive algorithm as well as benchmark in order to compare the behaviour of the closed loop system in this case with the behaviour of systems controlled by using the proposed adaptive and predictive algorithms. 3.2. An adaptive control algorithm Since the prior knowledge about the bioprocess supposed before is not realistic, in the following we develop an adaptive control strategy under the following conditions: the specific uptake rate ρ is incompletely known, since

ρm and Ks are considered unknown; the state variable X is not measurable; the on-line available measurements are S and Sin, but Sin

is time varying; the light intensity value is not set at this optimal value

Iopt and it is considered a strongly time varying function; the internal quota Q can be calculated, but this is not

used in the control design; all the other kinetic and process coefficients are known. Under these conditions an adaptive controller is obtained as follows. Since X is not measurable, and ρ is incompletely known, then in control law (4) the whole uptake rate uXS ϕ=ρ )( will be considered an unknown function that will be estimated by using an appropriately parameter estimator. Here we will use an observer-based parameter estimator (OBE) (for details see (Bernard, 2011; Dochain and Vanrolleghem, 2001; Petre et al., 2013; Petre and Selişteanu, 2013)). Since for this process we must to estimate only one completely unknown reaction rate, then using the dynamics of S, the proposed OBE is described by the following equations:

,)ˆ(ˆ

,)ˆ()(ˆˆ

SS

SSSSDS

u

inu

−γ−=ϕ

−ω−−+ϕ−=

(6)

where uϕ̂ is the on-line estimate of the unknown uϕ , and 0γ are tuning parameters to control the

stability and the tracking properties of the estimator. Usually, the values of these parameters are choosing by trial and error method. Then, the proposed adaptive control algorithm is obtained by combination of the parameter estimator equations (41)-(46) with the control law (26) rewritten as:

( ) ( )uin SSSSStD ϕ+−λ+⋅−= ˆ)(/1)( ** . (7) A block diagram of the proposed adaptive control system is shown in Fig. 1. 3.3. A nonlinear model predictive control

Now, a nonlinear model predictive control (NMPC) problem will be formulated for general case as follows. Fig. 1. Scheme of the adaptive closed-loop controlled system Consider the following discrete-time, time-invariant nonlinear system:

ξ=ξ=ξ +

),(),(1

kkk

kkkuhy

uf (8)

where kξ is the state vector, uk is the control input, yk is the process output, and f and h are nonlinear smooth functions. The control objective is to regulate the output variable yk to a specified setpoint value yref under certain state and input constraints:

.,

maxmin

maxmin

uuu kk

≤≤ξ≤ξ≤ξ

(9)

By using NMPC the solution of this constrained control problem is obtained by repeatedly solving the following optimization problem:

∑ Ω+∑ −Ψ−=

++=

++uN

iik

Tik

N

iikref

Tikref uuyyyy

11)()(min (10)

so that

≤≤ξ≤ξ≤ξ

ξ=ξ +

maxmin

maxmin

1 ),(

uuu

uf

k

k

kkk, (11)

where Ψ and Ω are two positive semidefinite matrices, N denotes the length of the prediction horizon and Nu the length of the control horizon. The model predictive control is a strategy that is based on the explicit use of some kind of system model to predict the controlled variables over a certain time horizon, called the prediction horizon (Eaton et al., 1990; Mayne et al., 2000). The NMPC control strategy used in this paper can be structured described as follows (Eaton et al., 1990): 1. A reference trajectory )( ktyref + , Nk ,,1= is defined which describes the desired system trajectory over the prediction horizon. 2. At each sampling time, the value of the controlled variable )( kty + is predicted over the prediction horizon

Nk ,,1= . This prediction depends on the future values of the control variable )( ktu + within a control horizon

uNk ,,1= .

ADAPTIVE CONTROLLER

UNCERTAIN BIOPROCESS

REACTION RATE

ESTIMATOR

S

Sin

D S*

_ +

uϕ̂



20

0 10 20 30 40 5094

96

98

100

102

104

106

108

110

112

Time (d)

Sin (

μmol

L−

1 )

0 10 20 30 40 50520

530

540

550

560

570

580

590

600

610

Time (d)

I (μ

E m

-2 s

-1)

3. The vector of future controls )( ktu + is computed such that an objective function (a function of the errors between the reference trajectory and the predicted output of the model) is minimised. 4. Once the minimisation is achieved, only the first optimised control action is applied to the plant and the plant outputs are measured. This measurement as well as the plant current state (either measured or estimated) is used as the initial states of the model to perform the next iteration (Șendrescu et al., 2011). Steps 1 to 4 that are repeated at each sampling instant are called a receding horizon strategy. The MPC based control strategy can be represented by the scheme shown in Fig. 2. Since usually a solution of the nonlinear least squares (NLS) minimization problem cannot be obtained analytically then it is computed by using numerical methods. There are many different methods of numerical optimization. To solve the NLS optimisation problem it was chosen the Levenberg-Marquardt (LM) algorithm. The LM algorithm is an iterative technique that locates the minimum of a multivariate function that is expressed as the sum of squares of non-linear real-valued functions (Kouvaritakis and Cannon, 2001; Wang and Boyd, 2008). It has become a standard technique for non-linear least-squares problems (Nocedal and Wright, 1999), widely adopted in a broad spectrum of disciplines. LM can be thought of as a combination of steepest descent and the Gauss-Newton method (Șendrescu et al., 2011). When the current solution is far from the correct one, the algorithm behaves like a steepest descent method. When the current solution is close to the correct solution, it becomes a Gauss-Newton method (Șendrescu et al., 2011). The previous general NMPC formulation is applied to microalgae process, in order to regulate the internal substrate S to a reference value S* by manipulating the dilution rate D under the same realistic conditions about the process as in the design of the adaptive control algorithm. The implementation of the presented predictive control strategies requires a discrete-time model of the process (1)-(4). This can be obtained by using, for example, the Euler approximation.

4. SIMULATION RESULTS AND DISCUSIONS

The behaviour and performance of the proposed adaptive and predictive control algorithms are examined by using simulation experiments in the case of the presented

NMPC Nonlinear bioprocess

Nonlinear model

Predicted output

Reference

Optimised input

Controlled output

Fig. 2. NMPC control strategy

microalgae process for which the bacterial growth rate is strongly nonlinear and time varying, the uptake rate is completely unknown, and the influent substrate concentration as well as the light intensity is strongly time-varying. The behaviour of these algorithms is compared to the exact linearizing control law (4) considered as benchmark. The simulations are achieved using the model (1)-(4) under identical circumstances. For the yield and kinetic coefficients are used the values given in the Table 1. Case 1: Adaptive control. The performance of the closed-loop system with the adaptive controller (7) is analysed, by comparison to linearizing controller (4), under the next conditions: the specific uptake rate ρ is incompletely known, that is

ρm and Ks are considered unknown; the state variable X is not measurable; the on-line available measurements are S and Sin, but Sin

is time varying as in Fig. 3; the light intensity value is not set at this optimal value

Iopt; it is assumed a strongly time varying function (that simulates some cloudy or rainy days and nights day), Fig. 4; Q is calculated, but this is not used in the control design; all the other kinetic and process coefficients are known. The behaviour of microalgae process in closed-loop system under these conditions is presented in Fig. 5-9. The graphics in Fig. 5 correspond to the controlled variable S, and Fig. 6 plots the control input D. To verify the regulation properties of control laws, a piece-wise constant variation was taken into consideration for *S . From Fig. 4 and Fig. 5 it can be observed that the substrate concentration S tracks the reference profile S*, and the control input D is maintained in the physical limits. Fig. 3. Time evolution of the influent substrate Sin Fig. 4. Time evolution of the light intensity I



21

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1

Time (d)

1 – Exact linearizing control 2 – Predictive control

Con

trol

inpu

t D

(d-

1 )

2

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (d)

1

2

1 – Exact linearizing control 2 – Adaptive control

Con

trol

inpu

t D

(d-

1 )

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

Con

trol

led

outp

ut S

(μm

ol L

-1) 1 – Exact linearizing control

2 – Adaptive control

*S

1

2

Time (d)

0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1


Time (d)

2

Con

trol

led

outp

ut S

(μm

ol L

-1)

*S

Fig. 5. Time evolution of output S – case 1 Fig. 6. Profile of control input D – case 1 Fig. 7. Profile of estimate of unknown parameter ϕu – case 1 Fig. 8. Profile of variable X – case 1 The gain of control laws (7) and (4) is =λ 20, and the tuning parameters of adaptive controller have been set to:

=ω -25, =γ 250.

The time evolution of the estimates of unmeasured variable uϕ provided by the OBE (6) is presented in Fig. 7. From this figure it can be noticed that the parameter estimator provide a very good result.

Fig. 9. Internal quota Q – case 2 In Figs. 8 and 9 are presented the time evolution of the biomass X and of the internal quote Q. From these figure one can observe that the evolution of the two variables are closed to the evolution of these variables in the benchmark case. Graphics in Figs. 5-9 show that the behaviour of the adaptive controlled system is correct, being very close to the behaviour of closed loop system in the ideal case (completely known process model) - benchmark case - when the exact linearizing controller (4) is used. The adaptive controller is able to maintain the controlled output S very close to its desired value, in spite of the unknowing of the substrate uptake rate and of the high variation of Sin and of the light I. Case 2: Predictive control. Now the performance of closed-loop system using the structure of predictive control law presented in Section 3.3 will be analysed, under the same realistic conditions about the process as in the case of the adaptive control system. The behaviour of closed-loop system using NMPC algorithm presented in Section 3.3 by comparison to the linearizing law (4) is presented in Figs. 10-14. Fig. 10. Time evolution of output S – case 2 Fig. 11. Profile of control input D – case 2

0 10 20 30 40 500

20

40

60

80

100

120

Time (d)

1

2

Rea

ctio

n ra

te ϕ

u (μ

mol

L−

1 s-

1 )

1 – Actual parameter ϕu 2 – Estimated parameter 2α̂

0 10 20 30 40 502

4

6

8

10

12

14

16


1

2

Time (d)

Var

iabl

e X (

μm3 L−

1 )

0 10 20 30 40 501

2

3

4

5

6

7

8

Time (d)

1

2

Inte

rnal

quo

ta Q

(μm

ol μ

m−

3 )




22

Fig. 12. Time variation of biomass X – case 2 Fig. 13. Internal quota Q – case 2 The parameters used for predictive control algorithm are defined, as follows: N = 6, Nu = 6, Ψ = I6, 601.0 I⋅=Ω , where I6 is the 6-dimensional unity matrix, and the time sampling is T = 10 min. Fig. 10 shows the time evolution of controlled output S, and Fig. 11 depicts the control input D. From Fig. 10 and Fig. 11 it can be observed that the substrate concentration S tracks the reference profile S*, and the control input D is maintained in the physical limits. The time evolution of the biomass X and of the internal quote Q are presented in Figs. 12 and 13. From these figure one can observe that the evolution of the two variables are closed to the evolution of these variables in the benchmark case. From graphics in Figs. 10-13 it can be seen that the behaviour of overall system with the predictive algorithm is correct, being very close to the behaviour of closed loop system with adaptive controller (7) presented in Case 1 as well as to the behaviour of closed loop system in the ideal case (exact linearizing controller). The predictive controller present good performance to lead the system to new set-points despite of strongly time-varying of the influent substrate concentration and of the light intensity.

5. CONCLUSIONS

In this paper an adaptive and a predictive control structures for a continuous microalgae process were designed and analysed. The two control strategies are developed under the realistic assumptions that the reaction rates are strongly nonlinear, time varying and incompletely known, and the influent substrate

concentration as well as the light intensity is strongly time-varying. The effectiveness of the proposed control laws was validated by numerical simulations. The adaptive control structure was achieved by combining a linearizing control law with a parameter estimator which plays the role of a software sensor for on-line estimation of uncertain or unknown bioprocess kinetic rates. The predictive control structure is based on Nonlinear Model Predictive Control (NMPC) strategy. The advantage of NMPC law is that it allows the current control input to be optimized, while taking into account the future system behaviour. This is achieved by optimizing the control profile over a finite time horizon, but applying only the current control input. The two proposed control strategies were tested in realistic simulation scenarios. Taking into account all the uncertainties and disturbances acting on the process, the conclusion is that the adaptive and predictive controllers can constitute a good option for the control of such class of bioprocesses.

ACKNOWLEDGMENT

This work was supported by UEFISCDI, project BIOCON no. PN-II-PT-PCCA-2013-4-0070, 269/2014.

REFERENCES

Angulo, F. Olivar G., and A. Rincon A. (2007). Control of an anaerobic upflow fixed bed bioreactor. In Proc. 15th Mediterranean Conf. on Contr. & Autom., July 27-29, 2007, Athens, Greece, Paper T04-005.

Bastin G. and Dochain D. (1990). On-line estimation and adaptive control of bioreactors. New York, NY: Elsevier.

Batstone D.J. et al. (2002). The IWA Anaerobic Digestion Model No 1 (ADM1). Water Sci. & Technology, vol. 45, no. 10, pp. 65-73.

Benattia S.E., Tebbani S., Dumur D., and Selisteanu D. (2014a). Robust nonlinear model predictive controller based on sensitivity analysis - Application to a continuous photo-bioreactor. 2014 IEEE Conf. on Control Applications (CCA), Part of 2014 IEEE Multi-conference on Systems & Contr., Oct. 8-10, 2014, Antibes, France, pp. 1705-1710.

Benattia S.E., Tebbani S., and Dumur D. (2014b). Nonlinear model predictive control for regulation of microalgae culture in a continuous photobioreactor. Proc. of the 22nd MED Conference, pp. 469–474.

Bernard O. (Responsible) (2004). Design of models for abnormal working conditions and destabilisation risk analysis. Report Number: D3.1b, TELEMAC IST 2000-28156, 81 pages.

Bernard O. (2011). Hurdles and challenges for modelling and control of microalgae for CO2 mitigation and biofuel production. J. Process Control, vol. 21, pp. 1378–1389.

Camacho E.F. and Bordons C. (2004). Model Predictive Control. London: Springer, 2004.

Dochain D. and Vanrolleghem P. (2001). Dynamical Modelling and Estimation in Wastewater Treatment Processes. IWA Publ., 2001.

Dochain D. Ed. (2008). Automatic Control of Bioprocesses. UK: ISTE / John Wiley & Sons.

Eaton J.W. and Rawlings J.R. (1990). Feedback control of nonlinear processes using online optimization techniques. Computers and Chemical Eng., vol. 14, pp. 469-479.

Time (d)

0 10 20 30 40 502

4

6

8

10

12

14

16

Var

iabl

e X (

μm3 L−

1 )


12

0 10 20 30 40 501

2

3

4

5

6

7

8

Inte

rnal

quo

ta Q

(μm

ol μ

m−

3 )

Time (h)

1 2




23

Hayakawa T., Haddad W.M., and Hovakimyan N. (2008). Neural network adaptive control for a class of nonlinear uncertain dynamical systems with asymptotic stability guarantees. IEEE Trans. N. Networks, vol. 19, pp. 80-89.

Ifrim G.A. et al. (2013). Multivariable feedback linearizing control of Chlamydomonas reinhardtii photoautotrophic growth process in a torus photobioreactor. Chemical Eng. Journal, vol. 218, pp. 191–203.

Kouvaritakis B. and Cannon M., Eds. (2001). Nonlinear Predictive Control: Theory and Practice. London, U.K.: The Institution of Electrical.

Logist F., Houska B., Diehl M., and Van Impe J.F. (2011). Robust multi-objective optimal control of uncertain (bio)chemical processes. Chem. Eng. Sci., vol. 66, no. 20, pp. 4670–4682.

Mairet F., Bernard O., Ras M., Lardon L., and Steyeret J.P. (2011). Modeling anaerobic digestion of microalgae using ADM1. Bioresource Technology, vol. 102, pp. 6823–6829.

Mayne D.Q., Rawlings J.B., Rao C.V., Scokaert P.O. (2000). Constrained model predictive control: stability and optimality. Automatica, vol. 36, pp. 789-814.

Neria-González I., Dominguez-Bocanegra A.R., Torres J., Maya-Yescas R., and Aguilar-Lópeza R. (2009). Linearizing control based on adaptive observer for anaerobic continuous sulphate reducing bioreactors with unknown kinetics. Chem. Biochem. Eng. Q., vol. 23, no. 2,

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